Exactly solvable nonlinear eigenvalue problems

Abstract

The nonlinear eigenvalue problem of a class of second order semi-transcendental differential equations is studied. A nonlinear eigenvalue is defined as the initial condition which gives rise a separatrix solution. A semi-transcendental equation can be integrated once to a first order nonlinear equation, e.g., the Ricatti equation. It is shown that the nonlinear eigenvalue problems of these semi-transcendental equations are equivalent to linear eigenvalue problems. They share the exactly same eigenvalues. The eigensolutions in the two problems are closely related. The nonlinear eigenvalue problem equivalent to the (half) harmonic oscillator in quantum mechanics is solved exactly. This is the first solvable nonlinear eigenvalue problem. The nonlinear eigenvalue problems of some extended equations are also studied.

First-order nonlinear eigenvalue problems involving functions of a general oscillatory behavior

“…) 3) to a Riccati equation, reference [8] introduces an exactly solvable nonlinear eigenvalue problem. That study is important because it presents a unique relationship between a class of nonlinear eigenvalue problems and corresponding linear ones, and it provides another justification for using terminology eigenfunctions and eigenvalues for nonlinear problems.…”